Most interesting and difficult problems in equilibrium statistical
mechanics concern models which exhibit phase transitions. For graduate
students and more experienced researchers this book provides an
invaluable reference source of approximate and exact solutions for a
comprehensive range of such models.
Part I contains background material on classical thermodynamics and
statistical mechanics, together with a classification and survey of
lattice models. The geometry of phase transitions is described and
scaling theory is used to introduce critical exponents and scaling laws.
An introduction is given to finite-size scaling, conformal invariance
and Schramm--Loewner evolution.
Part II contains accounts of classical mean-field methods. The parallels
between Landau expansions and catastrophe theory are discussed and
Ginzburg--Landau theory is introduced. The extension of mean-field
theory to higher-orders is explored using the Kikuchi--Hijmans--De Boer
hierarchy of approximations.
In Part III the use of algebraic, transformation and decoration methods
to obtain exact system information is considered. This is followed by an
account of the use of transfer matrices for the location of incipient
phase transitions in one-dimensionally infinite models and for exact
solutions for two-dimensionally infinite systems. The latter is applied
to a general analysis of eight-vertex models yielding as special cases
the two-dimensional Ising model and the six-vertex model. The treatment
of exact results ends with a discussion of dimer models.
In Part IV series methods and real-space renormalization group
transformations are discussed. The use of the De Neef--Enting
finite-lattice method is described in detail and applied to the
derivation of series for a number of model systems, in particular for
the Potts model. The use of Pad\e, differential and algebraic
approximants to locate and analyze second- and first-order transitions
is described. The realization of the ideas of scaling theory by the
renormalization group is presented together with treatments of various
approximation schemes including phenomenological renormalization.
Part V of the book contains a collection of mathematical appendices
intended to minimise the need to refer to other mathematical sources.
